Adhesive Categories

We introduce adhesive categories, which are categories with structure ensuring that pushouts along monomorphisms are well-behaved. Many types of graphical structures used in computer science are shown to be examples of adhesive categories. Double-pushout graph rewriting generalises well to rewriting on arbitrary adhesive categories.

The logic of linear functors

This paper describes a family of logics whose categorical semantics is based on functors with structure rather than on categories with structure. This allows the consideration of logics that contain possibly distinct logical subsystems whose interactions are mediated by functorial mappings. For example, within one unified framework, we shall be able to handle logics as diverse as modal logic, ordinary linear logic, and the ‘noncommutative logic’ of Abrusci and Ruet, a variant of linear logic that has both commutative and noncommutative connectives.Although this paper will not consider in depth the categorical basis of this approach to logic, preferring instead to emphasise the syntactic novelties that it generates in the logic, we shall focus on the particular case when the logics are based on a linear functor, in order to give a definite presentation of these ideas. However, it will be clear that this approach to logic has considerable generality.

A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on

Many problems lead to the consideration of “algebras”, given by an object A of a category A together with “actions” TkA → A on A of one or more endofunctors of A, subjected to equational axioms. Such problems include those of free monads and free monoids, of cocompleteness in categories of monads and of monoids, of orthogonal subcategories (= generalized sheaf-categories), of categories of continuous functors, and so on; apart from problems involving the algebras for their own sake.Desirable properties of the category of algebras - existence of free ones, cocompleteness, existence of adjoints to algebraic functors - all follow if this category can be proved reflective in some well-behaved category: for which we choose a certain comma-category T/AWe show that the reflexion exists and is given as the colimit of a simple transfinite sequence, if A is cocomplete and the Tk preserve either colimits or unions of suitably-long chains of subobjects.The article draws heavily on the work of earlier authors, unifies and simplifies this, and extends it to new problems. Moreover the reflectivity in T/A is stronger than any earlier result, and will be applied in forthcoming articles, in an enriched version, to the study of categories with structure.